Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

doi:10.1007/s42967-023-00279-5

Abstract

Abstract: In this paper, we propose a finite volume Hermite weighted essentially non-oscillatory (HWENO) method based on the dimension by dimension framework to solve hyperbolic conservation laws. It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears, and it is compact which will be good for giving the numerical boundary conditions. Furthermore, it avoids complicated least square procedure when we implement the genuine two dimensional (2D) finite volume HWENO reconstruction, and it can be regarded as a generalization of the one dimensional (1D) HWENO method. Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.

Key words: Finite volume, Dimension by dimension, HWENO, Hyperbolic conservation laws

Feng Zheng, Jianxian Qiu. Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 605-624.

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References

[1] Balsara, D.S., Altmann, C., Munz, C.-D., Dumbser, M.:A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG plus HWENO schemes. J. Comput. Phys. 226(1), 586-620 (2007)
[2] Capdeville, G.:A Hermite upwind WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 227(4), 2430-2454 (2008)
[3] Casper, J., Atkins, H.L.:A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems. J. Comput. Phys. 106(1), 62-76 (1993)
[4] Cockburn, B., Shu, C.-W.:The Runge-Kutta discontinuous Galerkin method for conservation laws V:multidimensional systems. J. Comput. Phys. 141(2), 199-224 (1998)
[5] Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.-D.:A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227(18), 8209-8253 (2008)
[6] Ha, Y., Gardner, C.L., Gelb, A., Shu, C.-W.:Numerical simulation of high Mach number astrophysical jets with radiative cooling. J. Sci. Comput. 24(1), 597-612 (2005)
[7] Hu, C.Q., Shu, C.-W.:Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150(1), 97-127 (1999)
[8] Hu, X.Y., Khoo, B.C.:An interface interaction method for compressible multifluids. J. Comput. Phys. 198(1), 35-64 (2004)
[9] Jiang, G.-S., Peng, D.:Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126-2143 (2000)
[10] Jiang, G.-S., Shu, C.-W.:Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202-228 (1995)
[11] Krivodonova, L., Xin, J., Remacle, J.F., Chevaugeon, N., Flaherty, J.E.:Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3), 323-338 (2004)
[12] Lax, P.D., Liu, X.-D.:Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19(2), 319-340 (1998)
[13] Liu, H., Qiu, J.:Finite difference Hermite WENO schemes for conservation laws, II:an alternative approach. J. Sci. Comput. 66(2), 598-624 (2016)
[14] Liu, X.-D., Osher, S., Chan, T.:Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200-212 (1994)
[15] Liu, Y., Zhang, Y.-T.:A robust reconstruction for unstructured WENO schemes. J. Sci. Comput. 54(2/3), 603-621 (2013)
[16] Luo, D., Huang, W., Qiu, J.:A hybrid LDG-HWENO scheme for KdV-type equations. J. Comput. Phys. 313, 754-774 (2016)
[17] Qiu, J., Shu, C.-W.:Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method:one-dimensional case. J. Comput. Phys. 193(1), 115-135 (2004)
[18] Qiu, J., Shu, C.-W.:Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II:two-dimensional case. Comput. Fluids 34(6), 642-663 (2005)
[19] Shi, J., Hu, C., Shu, C.-W.:A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175(1), 108-127 (2002)
[20] Shu, C.-W.:Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta. Numer. 29, 701-762 (2020)
[21] Shu, C.-W., Osher, S.:Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439-471 (1988)
[22] Tao, Z., Li, F., Qiu, J.:High-order central Hermite WENO schemes:dimension-by-dimension moment-based reconstructions. J. Comput. Phys. 318, 222-251 (2016)
[23] Titarev, V.A., Toro, E.F.:Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201(1), 238-260 (2004)
[24] Wang, C., Shu, C.-W., Han, W., Ning, J.:High resolution WENO simulation of 3D detonation waves. Combust. Flame 160(2), 447-462 (2013)
[25] Woodward, P., Colella, P.:The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115-173 (1984)
[26] Wu, K., Tang, H.:High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics. J. Comput. Phys. 298, 539-564 (2015)
[27] Wu, K., Yang, Z., Tang, H.:A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics. J. Comput. Phys. 264(1), 177-208 (2014)
[28] Xing, Y.L., Shu, C.-W.:High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208(1), 206-227 (2005)
[29] Xiong, T., Zhang, M., Zhang, Y.-T., Shu, C.-W.:Fast sweeping fifth order WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment. J. Sci. Comput. 45(1/2/3), 514-536 (2010)
[30] Zhang, Y.-T., Shu, C.-W.:Third order WENO scheme on three dimensional tetrahedral meshes. Communicat. Comput. Phys. 5(2/3/4), 836-848 (2009)
[31] Zhao, Z., Qiu, J.:A Hermite WENO scheme with artificial linear weights for hyperbolic conservation laws. J. Comput. Phys. 417, 109583 (2020)
[32] Zheng, F., Qiu, J.:Directly solving the Hamilton-Jacobi equations by Hermite WENO schemes. J. Comput. Phys. 307, 423-445 (2016)
[33] Zhu, J., Qiu, J.:A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes. Sci. China Ser. A 51(08), 1549-1560 (2008)
[34] Zhu, J., Qiu, J.:A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110-121 (2016)
[35] Zhu, J., Qiu, J.:A new type of finite volume WENO schemes for hyperbolic conservation laws. J. Sci. Comput. 73(2/3), 1338-1359 (2017)